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We are going to derive the Pythagorean Theorem from Heron's formula for the area of a triangle. This is not the best proof since it probably involves circular reasoning as most proofs of Heron's formula require either the Pythagorean Theorem or stronger results from trigonometry. For a more elementary proof, see Prove the Pythagorean Theorem.

Steps

1

Consider a right triangle ABC, right angled at vertex C, with sides as a, b and c where a and b are legs and c is the hypotenuse.

2

Our goal is to prove a2 + b2 = c2

3

The area can be written in two ways

Using Heron's Formula
Area = square root of s(s-a)(s-b)(s-c), where s is the semi-perimeter of the triangle given by (a + b + c) / 2

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wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. To create this article, 11 people, some anonymous, worked to edit and improve it over time. This article has also been viewed 15,957 times.